Solve each equation. Give solutions in exact form. See Examples 5–9. log₈ (x + 2) + log₈ (x + 4) = log₈ 8

검증된 단계별 안내

Recall the logarithm property that allows you to combine the sum of two logs with the same base: \(\log_b A + \log_b B = \log_b (A \times B)\). Apply this to the left side: \(\log_8 (x + 2) + \log_8 (x + 4) = \log_8 ((x + 2)(x + 4))\).

Rewrite the equation using the combined logarithm: \(\log_8 ((x + 2)(x + 4)) = \log_8 8\).

Since the logarithms on both sides have the same base and are equal, set their arguments equal: \((x + 2)(x + 4) = 8\).

Expand the left side: \(x^2 + 4x + 2x + 8 = 8\), which simplifies to \(x^2 + 6x + 8 = 8\).

Subtract 8 from both sides to set the quadratic equation to zero: \(x^2 + 6x + 8 - 8 = 0\), simplifying to \(x^2 + 6x = 0\). Then solve this quadratic equation for \(x\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Logarithms

Logarithmic properties, such as the product rule, allow combining or separating logarithms with the same base. For example, log_b(A) + log_b(B) = log_b(AB). This property is essential for simplifying the given equation by combining terms on one side.

Definition of Logarithms and Their Inverses

A logarithm log_b(A) answers the question: to what power must b be raised to get A? Understanding this helps convert logarithmic equations into exponential form, making it easier to solve for the variable.

Solving Equations and Checking for Extraneous Solutions

After manipulating the equation, solving for the variable involves algebraic techniques. Since logarithms are only defined for positive arguments, solutions must be checked to ensure they do not make any log argument non-positive, avoiding extraneous solutions.

Solve each equation. Give solutions in exact form. See Examples 5–9. log8 (x + 2) + log8 (x + 4) = log8 8